Let ax^{2} + bx
+ c = 0 and dx^{2} + ex + f = 0 have a common root a(say).
Then.

aa^{2} + ba + c = 0 and
da^{2}
+ ea + f = 0

Solving for a^{2} and a, we get

i.e. a^{2} = and
a
=

&⇒
(dc - af)^{2} = (bf - ce) (ae - bd)

which is the required condition for the two equations to have a common root.

Note:

Condition for both the roots to be common is

Find the conditions
on a, b, c, d such that equations 2ax^{3}+bx^{2}+cx +d =0 and
2ax^{2} + 3bx + 4c = 0 have a common root.

Let ‘a’ be a common root of the given two equations. .

Than 2aa^{3}+ba^{2}+ca +d = 0 …
(1)

and 2aa^{2} +3ba +4c = 0 …
(2)

Multiply (2) with a and then subtract (1) from it, we get

2ba^{2} +3ca - d =0 …
(3)

Now (2) and (3) are quadratic having a common root a, so

a^{2} =, a
= -

Eliminating a from these two equations we get,

(4bc + ad)^{2} =(bd
+4c^{2 })( b^{2}-ac) which is the required condition.